- Factor the denominator of the first fraction: \(x^2-16 = (x-4)(x+4)\).
- Express both fractions with the common denominator \((x-4)(x+4)\).
- Combine the numerators and simplify by distributing and grouping like terms: \(\frac{x-3(x+4)}{(x-4)(x+4)} = \frac{-2x-12}{(x-4)(x+4)}\).
- Factor out the common term in the numerator to get the final result: \(\boxed{\frac{-2(x+6)}{(x+4)(x-4)}}\).
Explanation
- Factor the Denominator
We are asked to simplify the expression \(\frac{x}{x^2-16} - \frac{3}{x-4}\). First, notice that the denominator in the first term, \(x^2-16\), is a difference of squares. We know the identity \(a^2 - b^2 = (a-b)(a+b)\), so we can factor \(x^2-16\) as \((x-4)(x+4)\). The expression becomes:
\[\frac{x}{(x-4)(x+4)} - \frac{3}{x-4}\]
- Finding a Common Denominator
To subtract these two fractions, we need a common denominator. The first fraction has the denominator \((x-4)(x+4)\), and the second fraction has the denominator \((x-4)\). The least common denominator is \((x-4)(x+4)\). We need to multiply the numerator and denominator of the second fraction by \((x+4)\) to match the denominator of the first fraction:
\[\frac{3}{x-4} = \frac{3(x+4)}{(x-4)(x+4)}\]
Now the expression is:
\[\frac{x}{(x-4)(x+4)} - \frac{3(x+4)}{(x-4)(x+4)}\]
- Combining and Simplifying the Fractions
Now that the denominators are the same, we can combine the numerators:
\[\frac{x - 3(x+4)}{(x-4)(x+4)}\]
Distribute the -3 in the numerator:
\[\frac{x - 3x - 12}{(x-4)(x+4)}\]
Combine the like terms in the numerator:
\[\frac{-2x - 12}{(x-4)(x+4)}\]
Factor out -2 from the numerator:
\[\frac{-2(x + 6)}{(x-4)(x+4)}\]
Comparing this result with the given choices, we see that it matches the second option.
Examples
Algebraic expressions involving rational functions appear frequently in engineering and physics, such as when analyzing electrical circuits or control systems. Simplifying these expressions is a fundamental skill that allows for a clearer understanding of how different components interact and how a system responds to changes. For example, if a signal is passed through multiple filters with different rational transfer functions, combining them into a single, simplified expression helps in calculating the overall system's gain and phase response. This process ensures that complex models can be computed efficiently and accurately.